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Mathematics Lessons for Remote LearningStrand: Geometry and Measurement Target: Y7, 8, 9, 10 –NZC Level 3/4Topic: All Strands – Thinking in Mathematics – For Teachers

Starter – Odd One Outlogical strategiccritical creative

The idea here is to select the odd one .

I choose __________ because _____________________KnowLearn by DoingTHE DEEP UNDERSTANDING OF THINKING

- Being enabled to know and understand, to connect concepts logically, to be creative and critical, and to communicate effectively in order to make sense of the world around us and solve problems encountered.

Thinking and MathematicsThinking is mathematics just as mathematics is thinking. “Cogito, ergo sum” This translates to “I think, therefore I am”. Rene Descarte wrote this and I see on this Wikipage that he wrote “dubito, ergo sum” as well. This translates to “I doubt, therefore I am!” See https://en.wikipedia.org/wiki/Cogito,_ergo_sum .

I like the “doubt” part. Being a skeptic is an essential part of modern day online information. Being able to sort the truth is vital. We do not have a knowledge issue in today’s world, we have a use of knowledge issue.

Early Thinking – NZC L1 – Age 0 to 6 (NZC = New Zealand Curriculum)Babies hear, feel and see certainly before birth, and afterwards very quickly learn or develop many ways of thinking. Very young children have very “me” oriented, “isolated” and not connected patterns of thought. Things happen without a reason. Magic. This is the reason the game of “Peck-a-Boo” works so convincingly. It is a very scary world and the they like to hold hands.

In young children we often see quite advanced conceptual development but there will be few connections and reasons. Being able to recite the counting numbers in order to 20 or more is not counting but good memory. In early thinking development observing, remembering, learning language is priority. I represent the thinking in NZC Level 1 as a group of disconnected circles. Each circle is an idea but there are no connections. There may be a few hazy dotted lines!

The job of the teacher is make more circles and start the process of joining them. This is done in reading by asking “So what do you think will happen now?” before turning the page. In mathematics touching the counters connects the count to the objects. Thankfully it soon becomes tedious for everyone however we do meet 14 year olds who count their way through quite complex problems.

Year 1 and 2 students can know many geometric facts and be able to measure. They like drawing and this should be encouraged. Experiences and using body movement to know concepts helps.

Thinking – NZC L2 – Ages 6, 7, 8 hopefully.The key barrier to more complex and connected thinking in NUMBER is the concept of PLACE VALUE. This is a complex concept, bathed in multiplication, and is much harder to understand that expected. Make haste slowly. When presented with the number 23 and the question to ask is... “Is there a 2 in this number?”. A knowing student will say “No” but there is a “20”.

Connections start to appear for grouping and using strategies such as “tidy numbers”. “Make ten”, “make 5”, and combining groups in clever ways starts to appear. This will all work on numbers up to about 30. When confronted with joining 37 and 9 there is a blur. Stick to simple connections because “connections” are what they are actually learning. The basic fact 3+7=10 is not about mathematics but about making ten, a strategy. Soon the connection to the related problem 3+8=?

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will be understood as well. At NZC L1 the student would simply see a new problem and count again, “8, 9, 10, 11”.

Each Level in the NZC was designed to take 2 years to “complete” What happens in practice is much more variable and students of any age can be at any level. The nzmaths.co.nz has research reports about this observation. I regularly measure about 60% of new Year 9 students at NZC L3. Very few at expectation of being at NZC L4, 5.

Additive Thinking – NZC L3 – Ages 9, 10, 11 and unfortunately olderThis level of thinking is much more connected. There are more strategies and there is more knowledge and there are many more skills and good motor control for use of mathematical equipment. These students choose addition every time however. They can also count.

These students are linear thinkers. This, then this, then this, then this. Typically these students will give an answer without a reason. The reason is not important, “I have the answer!”is all that is needed. Everything happens in a line or linear way. Connected but in 1 dimension.

These students can read, write and work with large numbers. All this using addition of course and they will know some multiplication ideas. They will derive an answer to “6x4 =?”. “How many legs on 6 horses”. Becomes “2 horses have 8 legs so 4 is 16 and 8 more makes 20 and 24. They have 24 legs”. If then asked what “4x6=” they will say “2x6 is 12 so 4 x 6 is 24”. Good connections.

In other strands, for example geometry, properties of triangles, angles adding to 180, 360 degrees in a circle, reading a compass for direction, following instructions, being right and wrong, making models, joining like terms in algebra, continuing linear patterns, using a protractor, are all available skills and knowledge. But never with a reason and always added.

The barrier of placevalue has been overcome and rich linear connections are available. Studying maths risks becoming boring but students are very obedient at this age. The best strategy for a teacher with these stduents is to get them out of there fast. You are doing the guiding! The sooner students enter the next level the better. Many adults revert to being a NZC L3 mathematician and never get to appreciate or enjoy a mathematical problem for the rest of their lives.

Multiplicative Thinking – NZC L4 – Ages 12,13,14 hopefully sooner. My first target for every student is to be measured at NZC L4. (See my website). This is an exciting complexity development of thinking. Students give answers and reasons. They know their times tables and use them. They understand what a formula is and even write units for measurements.

This is what I call two dimensional thinking. These students can also add and count efficiently. Being a multiplicative thinker is thinking of two things at once. 6x4 = 4x6 and 24 is recalled along with 24 has many factors and the multiples of 6 go on for ever. 1x24, 2x12, 3x8 and also 4,6. All this happens at the same time.

Students are still bound in the world of whole numbers but an increasingly complex understanding of fractions, decimals and percentage are being developed. Coonnections are made to the area or array model of multiplication. 2x2 is just as accessible as 2x2x2 and 2^5 now all make sense.

The diagram shows a more ordered description of being multiplicative. Only the array model should be used to describe multipplication. A nice diagnostic test is to draw a picture a 3x4=12. The only answer is in the diagram and shows the 3, the 4, the answer 12, the equals and the shape of multiplication. Likewise 1/3 x ¼ = 1/12 takes shape as well. What shape is multiplication? It is rectangular.

Being a multipicative thinker is an essential prerequisite for the next NZC Level 5. In all the strands this thinking is evidenced and is another good reason for teaching all strands. In Algebra generating an nth term is a good example. In geometry completing simple proofs and connecting ideas such as the angles of a quadrilateral sum to 360 because two triangles can be formed using a diagonal. In probability tossing a die and a coin can be calculated and used. In statistics more complex reasoning and connections to sample size and variation make sense. The world is a safer place.

At this level a student will answer a question and give a reason. They may even supply a couple of ways of solving a problem using different approaches. Being multiplicative is a huge step forwards and almost guarantees success in senior mathematics. Many professional people, builders, mechanics, nurses, teachers are effectively at this level. It is a powerful and useful place. But wait there’s more!

A diagnostic test. I buy 36 bales of hay at $4.50 each. How much do I pay altogether?

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There are many ways to do this problem and pondering a class of student answers will demonstrate the wide variation of creative thinking, strategic thinking and ways of recording already established even though the same course of study being undertaken. This demonstrates the messy way learning happens. You might think you know what you are teaching but you are wrong! You might think you know what they are learning but you are also wrong. Rest assured, they are learning! Something.

Multiplicative students start taking control of their thinking and can say “No”. Teachers should see the “No” as development not rebellion. The teacher just has to enrich the the learning programm and make it relevent for students to be engaged.

Proportional Thinking – NZC L5 and beyond – Ages 14+ but it could happen earlier.

“One does not become a proportional thinker by growing older!” The meaning here is that a student has to be taught to become a proportional thinker. Proportion is the real world of numbers and thought. Not a lot happens in simple ways in this world. The set of numbers at L5 becomes much more complex and includes all the fractions and decimals. Everything is connected and typically students will argue and reason quite well. Enjoy the change! Respect the development!

The diagram shows lines everywhere connecting all. These students have become quite logical, strategic, creative and critical. They can choose between all ways of thinking and self manage, persevere, make good choices, enjoy challenge and are actually a delight to teach.

That said, they need to be taught, challenged and rewarded. The learning environment has to be stimulating, relevent and meaningful.

The connections of all the previous learning is extended to fractions and decimals as a natural step and the world of mathematics can become a deep interest. It is also a powerful too for supporting learning in science and all other learning areas.

As a general guide every L5 student entering Year 11 will be an Excellent/Merit student in all subjects in Senior School. Knowing this, improving that success measure is an indication of sound learning and achievement.

Strand ThinkingNumber is pretty much described above. All forms of thinking apply to number. Number underpins all strands.

Algebra begins with understanding a variable and connecting this to a linear equation, the corresponding graph and how change in one part affects another. Another aspect of algebra is manipulation and solution and again connections to the visual graph and what solutions look like are powerful. Modern software exposes these as a normal expectation.

Measurement allows quantification of the dimensions and notions such as electric current. Measurement has deep connections to number and geometry. There is much logic in measurement and it is visual or sensed.

Geometric Thinking has a long history and is the place for visual proof. Logically stepping from one known place to another allows the proof of a huge consistent body of theorems. This is the world of Euclid.

Probability is a complex jungle of connected ideas involving number, geometry and measurement. This is the world of random. Making sense of random to generate unbiased samples, put limits of variation, estmate really difficult probabilities in the world of insurance and also be the key to unlocking QED and Q-Theory. Probability reigns supreme!

Statistics makes deep use of probability and then adds more twists and turns of complex relational thinking and language. The developments of statistical thinking using the PPDAC cycle of investigation make sense of “over time”, :two numerical variable relationships” and multivariate datasets with “comparative” questions. Today we have datasets with billions of data points only made sense of with high speed computation of the logical world of computers.

So, we all teach thinking when we teach mathematics. Keeping this mindset in front when we teach puts more emphasis on the thinking and less on the performance of mathematics.

Math AnxietyThere is much to be said for making every maths course an enjoyable experience and avoid building anxiety in learners through failure and marking problems wrong. The “tick” “cross” has a lot to answer for. Ask any group of adults about their mathematics experiences and use of mathematics. Here is what is Wikipedia says. https://en.wikipedia.org/wiki/Mathematical_anxiety

Geometry is making sense of size and shape

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There really is no reason to measure maths ability, just problem solving and thinking ability. Building perseverence, enjoyment of a challenge, being able to figure things all with “no speed clock” or “right-wrong measure” makes a lot of sense. Mathematics helps us all make sense of the world just as science does.

Everyone is entitled to have the opportunity of learning and enjoying mathematics.

Jo Boaler has a lot to say about this. https://www.youcubed.org/ Explore this website and look for the videos on Mindset. I agree with most of what Jo says. One thing I do not agree with her on is whether students should memorise multiplication tables and other basic facts. Of course they should Jo! It just saves a lot of time, develops memory and is a challenge that requires a bit of struggle. You recommend struggle!

JournallingToday I learned ________________________________________________________________

And I would like to know about ___________________________________________________

CommentsMake any comment you feel like making here.

Math Language: List all the math words you can find in this document and write what you think it means beside the word. Eg subtraction means to take away or to find the difference. Keeping a list of these words is a very good idea.

AnswersStarter – Odd One Out

logical strategiccritical creative

The idea here is to select the odd one .

I choose ____creative______ because _______that is where ideas come from________

FeedbackStudents and teachers are welcome to email jimhogan2@icloud.com with comments. This was a starter lesson that could be given to a NZC Level 2 or 3 student for some placevalue learning and revision. Students should select a set time each day and perhaps using the timer on a cell phone set 45 minutes or so to learn and practice mathematics. Keep trying on problems and expect to struggle. Persevering and struggling are great competencies to develop. You can learn more about these from https://www.youcubed.org/resource/growth-mindset/. We have a great math website in Nzwith a special resource called e-AKO https://nzmaths.co.nz/information-about-e-ako-pld-360 .

Geometry is making sense of size and shape